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H: prove that $\frac{b+c}{a^2}+\frac{c+a}{b^2}+\frac{a+b}{c^2}\ge\frac{1}{a}+\frac{1}{b}+\frac{1}{c}$
Assume: $a,b,c >0$ prove that :
$$\frac{b+c}{a^2}+\frac{c+a}{b^2}+\frac{a+b}{c^2}\ge\frac{1}{a}+\frac{1}{b}+\frac{1}{c}$$
AI: Replace $(a,b,c)$ by $(x_1,x_2,x_3)$ for notational convenience, and start with the inequal... |
H: Finding an accurate rating between 1 and 10 based on numerous data points
I'm going to provide a fictitious setup that aligns with my mathematical needs. I am a company that is measuring the bounciness of balls and providing two 1-10 rating to each ball based on how it compares to all other balls. The first rating ... |
H: Axiomatization of $\mathbb{Z}$ via well-ordering of positives.
Though I've seen several cool axiomizations of $\mathbb{R}$, I've never seen any at all for $\mathbb{Z}$.
My initial guess was that $\mathbb{Z}$ would be a ordered ring which is "weakly" well-ordered in the sense that any subset with a lower bound has a... |
H: Does this equation imply the Class Equation?
At this time, I am reading the following theorem.
Let $G$ be a group acting transitively on a set $\Omega$. Then
\begin{equation}
|G|=\sum_{g\in G}\chi(g) \tag{$\clubsuit$}
\end{equation}
wherein $\chi(g)=|\{\alpha \in \Omega :\alpha^g=\alpha\}|$.
It seems to me th... |
H: Applying a contraction to balls' centers increases the size of the balls' intersection?
The following statement seems clearly true, but I'm having a hard time proving it:
Fix $\alpha\in[0,1]$. Let $\mu$ be Lebesgue measure. For $r\ge 0$, let $B(c,r)\equiv[c-r,c+r]$. Fix $r_1,\ldots,r_n\in[0,\infty)$, and $c_1,\ldot... |
H: Checking divisibility by large numbers
I am currently familiar with the method of checking if a number is divisible by $2$, $3$, $4$, $5$, $6$, $8$, $9$, $10$, $11$. However is there a way to check if a no is divisible by $23$ ?
I read something at this link regarding this matter but couldn't really figure out wha... |
H: What does sup mean?
I found this formula regarding calories burned:
Rate per Pound (Cal/lb-min)=A+BV+CV.sup.2 +KDV.sup.3 where:
V=Running Speed (mph)--limited to a minimum of 3 mph and a maximum of
14 mph
A=0.0395
B=0.00327
C=0.000455
D=[0.00801(W/154).sup.0.425 ]/W
W=Weight (lbs)
K=0 or 1 (0=Treadmill; 1... |
H: Maximum point of a polar function
I have a curve C with polar equation
$$r^2 = a^2\cos{2\theta} $$
And I am looking to find the length $x$ when $r=max$
Judging from the equation:
$$r = \sqrt{a^2\cos{2\theta}} $$
R will be maximum at $\cos{2\theta}=1$
So the maximum value of $r$ is:
$$r = \sqrt{a^2} =a$$
However t... |
H: game show problem
Possible Duplicate:
The Monty Hall problem
You are in a game show. You have to choose between three buttons, A, B and C. Pressing one of them will give you £200,000, and pressing either of the other two will give you a free mousemat. You choose a button at random (button A). The gameshow host d... |
H: Are Different Areas of Number Theory That Different
I was wondering if number theorists are "number theorists," or eventually resolve themselves into one of the various branches - i.e., algebraic, analytic, etc.
Also out of curiosity, I was wondering if there is a particular branch or area that draws the most atten... |
H: Simplifying the expression of exponential and logarithms
I want to simplify the following expression.
$$Y=\text{Bottom} + \frac{\text{Top}-\text{Bottom}}{1+10^{((\log EC50-X))}}$$
$\log$ is base of $10$. Some may know that it's a dose response curve, and I want to solve for $EC50$.
I tried simplifying it but I forg... |
H: "Mixing" the diagonals of a positive semidefinite matrix
I have arrived at this result from a very different perspective (quantum operations) but, being a completely algebraic result, I was hoping that there would be a simple algebraic way of looking at it too.
Let $P$ be a positive semidefinite matrix. Let $E$ be ... |
H: convert values from one coordinate system (x,y) to another coordinate system (x', y')
Following is a graph that contains both coordinate systems (x,y) and (x',y'). x, y, x', and y' are all axes
y'
^
|
+----------> x
| | | |
+--+--+--+
| | | |
+--+--+--+
| | | |
+--+--+--+-> x'
|
v
y
Above is a picture of ... |
H: Divisibility criteria of 24. Why is this?
I am currently familiar with the method of checking if a number is divisible by $2, 3, 4, 5, 6, 8, 9, 10, 11$. While Checking for divisibility for $24$ (online). I found out that the number has to satisfy the divisibility criteria of $3$ and $8$. I agree this gives the answ... |
H: If $k \equiv 1 \pmod 4$ and $q > 3$, does it follow that ${q^k}\sigma(q^k) \equiv q\sigma(q) \equiv 2 \pmod {q - 1}$?
If $k \equiv 1 \pmod 4$ and $q > 3$ (where $q$ is prime), does it follow that ${q^k}\sigma(q^k) \equiv q\sigma(q) \equiv 2 \pmod {q - 1}$?
Observe that $q\sigma(q) \mid {q^k}\sigma(q^k)$ when $k \eq... |
H: Vector as argument of a function
Given a function $f(x)=y$ is correct to say that $f\left(\left[\begin{array}{c} x_1 \\ x_2 \\ x_3 \end{array}\right]\right)=\left[\begin{array}{c} y_1 \\ y_2 \\ y_3 \end{array}\right]$?
AI: Yes, assuming that $x \in A^3$ and $y \in B^3$ for some suitable spaces $A,B$ you just give t... |
H: Why are the solutions to second order differential equations summed?
I'm an A-Level maths student, and earlier in the year we learnt about second order differential equations in the form $ A\frac{d^2y}{dx^2} + B \frac{dy}{dx} + Cy = 0$ and $ A\frac{d^2y}{dx^2} + B \frac{dy}{dx} + Cy = f(x)$
For the first form, we l... |
H: Group structure on $\mathbb R P^n$
For which positive integers $n$ can $\mathbb R P^n$ be given the structure of a topological group?
I believe that $\mathbb R P^n$ cannot be given a Lie group structure for even $n$, since then it is not orientable. But, this doesn't necessarily imply it doesn't have a topological... |
H: Finding the matrix of multiplication by $\theta^2$, where $\theta^3 - 3\theta + 1 = 0$
This is a problem from a on-line source which yet comes with a solution (self-studier; not h.w.).
Let $E = \mathbb Q(\theta)$, where $\theta$ is a root of the irreducible polynomial
\[
X^3 -3X + 1.
\]
I was wondering how one gets... |
H: ODE series solution with IVP
I am a little rusty with initial conditions. If anyone can confirm that I nailed this then I would be very happy. i need to write about five terms in my series to the solution of
$xy'' + y = 0$
$y(1) = 5$
$y'(1) = 0$
So my solution will look something like $y = \sum_{n=0}^{\infty}a_n (x... |
H: $U\subset{R}^m$ open $f\colon U\to N$ local homeo and $y\in N$ with $\operatorname{card}\big(f^{-1}(\{y\})\big)$ is infinite then $f$ is not closed.
Let $U$ be an open set. If $f\colon U\subset{R}^m\to N$ is a local homeomorphism and exists $y\in N$ such that $\operatorname{card}\big(f^{-1}(\{y\})\big)$ is infinit... |
H: Cardinality of varieties
Does a (connected?) projective (or just complete?) variety over a finite field have cardinality congruent to 1 mod the size of the field?
Do (connected?) affine varieties have cardinality a power of the size of the field?
I think the answers to these questions are supposed to be yes, but I ... |
H: Eigenvalue For A 2-Fold Tensor
This is problem 4 from page 258 of Curtis's Linear Algebra: An Introductory Approach. I seem to be having trouble understanding something needed to solve the problem, which reads
Suppose A and B are matrices in triangular form, with zeros
above the diagonal. Show that A $\times$ ... |
H: Finding the Number of Ordered Triples $(x_1,x_2,x_3)$ such that $x_1 + 2x_2 + 3x_3 = n$
I would like to find the number of ordered triples $(x_1,x_2,x_3)$ such that $x_1 + 2x_2 + 3x_3 = n$. For each $n$ call this number $r(n)$. So after reading some similar questions on this site I'm pretty sure this is just some... |
H: Looking for some simple topology spaces such that $nw(X)\le\omega$ and $|X|>2^\omega$
I believe there are some topology spaces which satisfying the network weight is less than $\omega$, and its cardinality is more than $2^\omega$ (not equal to $2^\omega$), even much larger.
Network: a family $N$ of subsets of a to... |
H: Isomorphism between quotient modules
Is it true for a commutative ring $R$ and its ideals $I$ and $J$ that if the quotient $R$-modules $R/I$ and $R/J$ are isomorphic then $I=J$?
AI: No, just take a polynomial ring $k[X]$ for some field $k$ (which is a commutative domain), and consider the ideal generated by $X$ and... |
H: Given Linear function, find function describing decrease in the rate of change x vs y
Given a linear function such as
$y = 1.62*x - 0.49$
Scenario 1
If $x = .5$ then $y = .32$
If we then increase $x$ by $10$% ($x=.55$), then $y=.401$, which results in $y$ increasing by $\approx .25$%
Scenario 2
If $x = .6$ then $y ... |
H: Compact resolvent VS certain boundedness condition
The following question is motivated by the definition of spectral triples in noncommutative geometry. This question was split in the following parts:
First: Could somebody give diverse examples of operators on Hilbert spaces, having compact resolvent?
Now, suppo... |
H: Injective linear map between modules
If you have an injective linear map between two free modules of equal dimension, is the determinant of the matrix representing the map necessarily nonzero? If not is there an obvious counterexample? (Everything is over a multivariate polynomial ring over a field.)
Thanks!
AI: ... |
H: Eckmann-Hilton and higher homotopy groups
How does the Eckmann-Hilton argument show that higher homotopy groups are commutative?
I can easily follow the proof on Wikipedia, but I have no good mental picture of the higher homotopy groups, and I can't see how to apply it. Wikipedia mentions this application in one... |
H: Prove $F(x,y,z)=o(||(x,y,z)||)(x,y,z)$ has a vector potential
This is an unsolved exercise given in my textbook, which I am having trouble with. The exercise seems simple, but for some reason I can't solve it. Help would be very nice!
Let $o(t)$ be a real continuous positive function in $[0,\infty)$, and $F(x,y,z)... |
H: Applying Dominated Convergence and/or Monotone Convergence for $|Z|^{1/n}$
Problem:
I am self-learning about DCT and MCT and related Lemma's. I understand the theorems as constructed, but I am struggling to apply them.
As an example:
$X_n=|Z|^{1/n}$ where $Z\sim N(0,1)$
I am trying to do the following:
a)Identify $... |
H: Group of arbitrary order
Can we construct a group of order $n$ for any $n \in \mathbb{Z}^+$ i.e set of positive integers? Are there theorems which characterize the order of any finite group? What is the smallest possible restriction you can have on a group such that there does not exist such a group of particular o... |
H: Discriminant for $x^n+bx+c$
The ratio of the unsigned coefficients for the discriminants of $x^n+bx+c$ for $n=2$ to $5$ follow a simple pattern:
$$\left (\frac{2^2}{1^1},\frac{3^3}{2^2},\frac{4^4}{3^3},\frac{5^5}{4^4} \right )=\left ( \frac{4}{1},\frac{27}{4},\frac{256}{27},\frac{3125}{256} \right )$$
corresponding... |
H: Does an injective endomorphism of a finitely-generated free R-module have nonzero determinant?
Alternately, let $M$ be an $n \times n$ matrix with entries in a commutative ring $R$. If $M$ has trivial kernel, is it true that $\det(M) \neq 0$?
This math.SE question deals with the case that $R$ is a polynomial ring... |
H: Why is it legit to evaluate $\lim_{x\rightarrow 1} \frac{(x-1)(x+1)}{x-1}$ by cancelling common factors?
I haven't ever taken an analysis course, so maybe that's where I would really learn this, but I've always wondered why it's okay to do this when evaluating a limit. I guess it's the case that there is a theorem ... |
H: Prove that: $\frac1{20}\le \int_{1}^{\sqrt 2} \frac{\ln x}{\ln^2x+1} dx$
I'm interested in proving the following integral inequality:
$$\frac1{20}\le \int_{1}^{\sqrt 2} \frac{\ln x}{\ln^2x+1} dx$$
According to W|A the result of this integral isn't pretty nice, and involves the exponential integral.
AI: Let $f(x) = ... |
H: What is the total number of combinations of 5 items together when there are no duplicates?
I have 5 categories - A, B, C, D & E.
I want to basically create groups that reflect every single combination of these categories without there being duplicates.
So groups would look like this:
A
B
C
D
E
A, B
A, C
A, D
A, E
... |
H: group homomorphism from $S^1\times S^1$ to itself
Could any one give me a hint how to show that if the kernel of a group homomorphism from $S^1\times S^1$ to itself is finite then it must be cyclic subgroup of $S^1\times S^1$?
AI: Denote the kernel $K=\ker\varphi$ of a homomorphism of $S^1\times S^1$ to itself. S... |
H: How to find large prime factors without using computer?
What is the largest prime factor of the number 600851475143 ?
This is the third problem of Project Euler.
How to approach this mathematically (without computer programming) ?
AI: Well the point of Project Euler is to program. This is a problem that you could... |
H: What reference contains the proof of the classification of the wallpaper groups?
Background: I am doing a course on Groups and Geometry ( Open University M336 ). One of the topics is the classification of the plane symmetries, a.k.a. The Wallpaper Groups.
Question: What reference contains the original proof that th... |
H: Lebesgue Integral, existence, improper integrals, etc.
Problem:
At the request of another user, I am taking an older question and specifically addressing one problem.
I am self-learning about Lebesgue integration, and am just starting to try and apply some examples of the existence of the integral.
For the function... |
H: Compute: $\lim_{n\to\infty} \{ (\sqrt2+1)^{2n} \}$
Compute the following limit:
$$\lim_{n\to\infty} \{ (\sqrt2+1)^{2n} \}$$
where $\{x\}$ is the fractional part of $x$. I need some hints here. Thanks.
AI: Consider $$ (\sqrt2+1)^{2n} + (\sqrt2-1)^{2n} $$
Try to show that it is an integer and hence this fractional... |
H: Find all primes $p$ such that $(2^{p-1}-1)/p$ is a perfect square
Find all primes $p$ such that $(2^{p-1}-1)/p$ is a perfect square. I tried brute-force method and tried to find some pattern. I got $p=3,7$ as solutions. Apart from these I have tried for many other primes but couldn't find any other such prime.
Are ... |
H: Definition of S(n) for graded ring S
In Hartshorne's Algebraic Geometry, the twisting sheaf of Serre $\mathscr{O}(n)$ is defined to be $S(n)^\sim$, where $S$ is an $\mathbb{N}$-graded ring. But I couldn't find the definition of $S(n)$ anywhere in the book (if any one knows where it is, please let me know). It is pr... |
H: Expectation of a minimum
Why is this: $E(\tau \wedge T) = \int_0^Ttf_\tau(t)dt+T(1-P(\tau\leq T))$, where $f_\tau(t)$ is the pdf of $\tau$. I am thinking its because $E(\tau \wedge T) = \tau P(\tau\leq T)+T(1-P(\tau\leq T))$ since $\tau \wedge T =\min(\tau,T) = \tau$ if $\tau\leq T)$ and T if $\tau> T)$. But the fi... |
H: Similar Matrices and Equivalence Relations
Since similarity of matrices' is an equivalence relation, doesn't that imply that given any polynomial equation involving similar matrices you can substitute in any similar matrices' and the equation will still hold?
For example, given $A,B,C\in M^F_{n\times n}$
if $B \co... |
H: Calculate: $\lim_{n\to\infty} \int_{0}^{\pi/2}\frac{1}{1+x\tan^{n} x }dx$
I'm supposed to work out the following limit:
$$\lim_{n\to\infty} \int_{0}^{\pi/2}\frac{1}{1+x \left( \tan x \right)^{n} }dx$$
I'm searching for some resonable solutions. Any hint, suggestion is very welcome. Thanks.
AI: Note that the integra... |
H: Extension of Yau's theorem to general bundles
Calabi-Yau manifolds have the nice property that $c_1(TM) = 0$ implies there is a Ricci flat metric: $\text{Ric}(\omega)$.
Is it possible to construct a similar theorem vor a Vector Bundle over a Calabi-Yau manifold?
i.e. $c_1(V) = 0$ implies that there exists some fla... |
H: An elementary congruence question in number theory
If ${q^k}\sigma(q^k) \equiv a \pmod b$ and $\displaystyle\frac{\sigma(q^k)}{n} \neq \displaystyle\frac{\sigma(n)}{q^k}$, does it follow that $n\sigma(n) \not\equiv a \pmod b$?
Here, $q$ is prime and $n$ is composite.
AI: We can generate counterexamples at will. Ta... |
H: Subtraction of numbers with arbitrary bases
Possible Duplicate:
How to do +, -, *, / with number in a base b?
I am reading research papers in the category of recreational mathematics on the topic of numbers similar to Kaprekar number. Almost all the time we come across subtration of numbers with arbritrary bases... |
H: Solving $ \frac{3x+3}{\sqrt{x}}=4+\frac{x+1}{\sqrt{x^{2}-x+1}} $
Solve in $\mathbb{R}$:
$$ \frac{3x+3}{\sqrt{x}}=4+\frac{x+1}{\sqrt{x^{2}-x+1}} $$
AI: Note that $\frac{3x+3}{\sqrt{x}}\ge 6$, with equality only at $x=1$. This comes down to the inequality $(\sqrt{x}-1)^2 \ge 0$.
Or else we can simply quote the fact... |
H: Density character of a subspace of a topological space.
Let $(X,\tau)$ be a topological space. Suppose $dc(X)=\kappa$ and let $D\subset_{dense} X$ be a dense subset of $X$ of cardinality $\kappa$. Is it true that $X\setminus D$ has density character $\kappa$, as a subspace of $X$ with the restricted topology?
AI: N... |
H: Find the limit of: $\lim_{n\to\infty} \frac{1}{\sqrt[n+1]{(n+1)!} - \sqrt[n]{(n)!}}$
Could be the following limit computed without using Stirling's approximation formula?
$$\lim_{n\to\infty} \frac{1}{\sqrt[n+1]{(n+1)!} - \sqrt[n]{(n)!}}$$
I know that the limit is $e$, but I'm looking for some alternative ways that ... |
H: What is $\overline{\sin({z})} $ equal to?
What is
$\overline{\sin(z)}$
equal to?
AI: I will write $\exp(x)$ instead of $e^{x}$, they are synonyms. (Just a notational warning!)
Well, recall Euler's formula
$$ \exp(i\theta)=\cos(\theta)+i\cdot\sin(\theta).$$
Then we see
$$\exp(i\theta)-\exp(-i\theta)=2i\cdot\sin(\the... |
H: How to prove $\frac{4^{1/\log_4(3/4)}}{3^{1/\log_3(3/4)}} = \frac{1}{12}\ ?$
How could we prove that
$$ \frac{4^{1/\log_4(3/4)}}{3^{1/\log_3(3/4)}} = \frac{1}{12}\ ?$$
I have reduced it the form
$$\frac{4^{\ln(4)/\ln(3/4)}}{3^{\ln(3)/\ln(3/4)}}$$
I am not sure what to do next to get snappy solution. Any ideas?
AI:... |
H: How can I convert between powers for different number bases?
I am writing a program to convert between megabits per second and mebibits per second; A user would enter 1 Mebibits p/s and get 1.05 Megabits p/s as the output.
These are two units of computer data transfer rate. A megabit (SI unit of measurement in deny... |
H: Borel regular measure
I stuck with this question, can you help me please.
Is it exist $ \mu$ - Borel regular measure in $[0,1]$ so that to all polynomial $p$ one has:
$\int_{[0,1]}p(t)d \mu(t)=p'(0)$?
Thanks a lot!
AI: For such a measure $\mu$ we have, for $n=0$ and $n\geq2$,
$$\int_{[0,1]}t^n d \mu(t)=0$$ and hen... |
H: A computation in a Hilbert space
Can someone give me an idea, why $\forall x: \left<\sum_j \lambda_j \left< x,e_j\right> e_j,x\right>\geq 0$, where the $\lambda_j$'s are fixed, implies that all $\lambda_j$ are $\geq0$,?
(The $x$'s belong to a Hilbert space,the $e_j$'s are an orthonormal basis and the $\lambda_j$'s... |
H: Combination of n sets that produces a set of n-tuple
Given n sets with $3$ elements:
$X_i=\{a_i,b_i,c_i\}$
where $\{i\in\mathbb{N}\ |\ 1\leq i\leq n\}$. How can I define a n-tuple based on combination of this sets that produces the set $S$ with $3^n$ elements ($n$-tuples) as following:
$S=\{(a_1,a_2,\cdots,a_n),(... |
H: Computing the derivative from the definition
Using the limit definition of the derivative which I know is:
$$f'(x)=\lim_{h\to0}\left(\frac{f(x+h)-f(x)}{h}\right)$$
I am trying to solve this problem
$$f(x)= \frac{x}{x+2} $$
How do I go about properly solving this, I seemed to get
$$\frac{x}{x+2}\ $$
as my ... |
H: Help with general integration problem
I have a specific problem that Ive generalized here for simplicity.
Let $F(x)=\int^{g(x)}_0h(x,y)dy $
Suppose $F(0)=0$ (with $g(0)>0$)
Now suppose that $h$ is increasing in $y$. Then, it follows that:
$F(x) \leq g(x) h(x,g(x)) $
Does it therefore have to be the case that $h(0,g... |
H: infinity understanding problem?
between 0 meter -> 1 meter
there are 100 cm.
but each cm has infinite numbers :
for example between 0..1 cm there are :
0.000000000001
..
0.00000000000111
..
0.000000000001111111
and more numbers and combinations...
..
..
..
1.0
to each number I c... |
H: How do you Compute a Squareroot limit?
Possible Duplicate:
Limits: How to evaluate $\lim\limits_{x\rightarrow \infty}\sqrt\[n\]{x^{n}+a_{n-1}x^{n-1}+\cdots+a_{0}}-x$
I am trying to compute the limit here, i am not sure how to do it if i just plug it in or do i do the conjugate and solve? For some reason i got in... |
H: How is this expression simplified?
I have this:
$$\sqrt{(dx)^2 + (dy)^2}$$
And my book simplified it as:
$$\sqrt{1 + \Big(\frac{dy}{dx}\Big)^2} \times dx$$
I don't have even a close idea how he did it. If it helps, is about path lenght whit integration.
AI: $$a\sqrt{r} = \sqrt{a^2(r)}\quad\text{if }a\gt 0\text{ and... |
H: Differential equations and family of function solutions
I can't follow what Stewart is doing in his book. I can easily follow his work but his conclusion doesn't make any sense to me.
"Show that ever member of the family of functions
$$y = \frac{1+ce^t}{1 - ce^t}$$ is a solution of the differential equation $$y' = ... |
H: Teaching abstract maths concepts to young children.
I am interested in opinions and, if possible, references for published research, about the pros and cons of teaching abstract maths concepts to young children. My younger brother (five years old) understands negative numbers and square roots so I was thinking of t... |
H: Gaussian curvature in $S^3$
I'm trying to read a survey paper on the Willmore conjecture and I'm missing a lot of basic knowledge. In particular, let $u: \mathcal{M} \rightarrow S^3 \rightarrow \mathbb{R}^4$ be a smooth immersion of a compact orientable two dimensional surface into the standard 3-sphere, and let $\... |
H: The pointwise liminf of a sequence of upper semicontinuous functions is upper semicontinuous. Find the flaw in my counterexample?
Let $f_n: \mathbb{R} \to \mathbb{R}$ be defined as follows:
$f_n$ is even.
$f_n(0) = \frac{1}{2}$
$f_n(x) = 0$ if $0<x< \frac{1}{n}$
$f_n(x) = 1$ if $x> 2/n$
$f_n$ is linear on $(\frac{1... |
H: Probability of sequence being longer than some length
We are given random bit generator which generates 0s and 1s with equal probability 1/2. We have an algorithm which generates random numbers using this random bit generator in this way: we look for subsequences of 1s, and length of each subsequence gives us one r... |
H: Limit to Infinity
I have a question to compute a limit, i understand that the limit is infinity but How can i show the solution to get this answer to infinity. I understand that it will be large pos #'s on top and small #'s on bottom making it go to infinity but how can i write it out to prove this? At least how wo... |
H: Prove that: $\int_{0}^{1} \frac{x^{4}\log x}{x^2-1}\le \frac{1}{8}$
Here is another interesting integral inequality :
$$\int_{0}^{1} \frac{x^{4}\log x}{x^2-1}\le \frac{1}{8}$$
According to W|A the difference between RS and LS is extremely small, namely 0.00241056. I don't know what would work here since the differe... |
H: Are Tonelli's and Fubini's theorem equivalent?
I can derive Fubini's theorem for interated integrals of complex functions from Tonelli's theorem for iterated integrals for unsigned functions. I was wondering whether there is a way to go backwards. I do not think so, because Fubini's theorem assumes the integrals ... |
H: Are there any conditions for $G$ until above action has non-trivial kernel?
Let $G$ is a group and $H$ be a subgroup of it. Then $G$ can act on the following set $$\Omega= \{Hg|g\in G\}$$
by $\forall Hg\in\Omega$ and $x\in G$; $(Hg)^x=Hgx$ (I don't know if I can call this action right regular representation of $G$?... |
H: Proving that $|d|$ is not a prime number
Assume $a,b,c$ are integers and $a+b+c=0$.
If $d=a^{1433}+b^{1433}+c^{1433},$ prove that $|d|$ is not a prime number.
AI: Note that $1433$ is prime. It follows by Fermat's theorem that $a^{1433}\equiv a\pmod{1433}$, with similar results for $b$ and $c$. Thus
$$d=a^{1433}+b^{... |
H: Tuple definition
Is it correct?
$S=\{\langle t,h\rangle:t\in\{0,\Delta t,2\Delta t,\cdots,24\},h\in\{0,\Delta h,2\Delta h,\cdots,H\}\}$
I would like to say that $S$ is a 2-tuple. The first tuple can vary from $0$ to $24$, with $\Delta T$ step, and the second one can vary from $0$ to $H$, with $\Delta H$ step.
AI: $... |
H: What's an intuitive explanation of the max-flow min-cut theorem?
I'm about to read the proof of the max-flow min-cut theorem that helps solve the maximum network flow problem. Could someone please suggest an intuitive way to understand the theorem?
AI: Imagine a complex pipeline with a common source and common sink... |
H: Uncountability of $\overline{\mathbb{F}_p}$.
In the following MathOverflow question, it has been pointed out that $\overline{\mathbb{F}_p}$ is an uncountable set. Whereas according to http://press.princeton.edu/chapters/s9103.pdf (see page 4 theorem 1.2.1) the closure $\overline{\mathbb{F}_p}$ is $\cup_{n=1}^{\inft... |
H: Simplifying $\frac{2^{n + 4} + 2^{n + 2} + 2^{n - 1}}{2^{n - 2} + 2^{n - 1}}$
I'm stuck in the follow equation:
$$\dfrac{2^{n + 4} + 2^{n + 2} + 2^{n - 1}}{2^{n - 2} + 2^{n - 1}}$$
As all the bases are equal, I got $\dfrac{3n + 5}{2n - 3}$
Where I've to go now ?
Thanks
EDIT:
Then, my initial idea was totally wrong,... |
H: Doubt on displacement of a parabola(Again)
In another exercise is given:
Find the parabola which is a displacement of $y = 2x^2 - 3x + 4$ which passes though the point $(2, -1)$ and has $x = 1$ as its symmetry axis.
I've reduced the based equation to the form: $y = 2(x - \frac{3}{4})^2 + \frac{23}{8}$, so the verte... |
H: Convex cone of functions of the form $a\log(1+bx + ?)$
Given the set of all functions
\begin{align}
f_{a,b}(x): [0, \infty) &\to [0, \infty) \\
x &\mapsto a\log(1+bx)
\end{align}
Is there a way of making the set of these functions $S =\{f_{a,b}: a,b \ge 0\}$ have the property that $\left[f_{a,b} + f_{a',b'}\right] ... |
H: Integrating with respect to different variables
I have started reading a book on differential equations and it says something like:
$$\frac{dx}{x} = k \, dt$$
Integrating both sides gives
$$\log x = kt + c$$
How is it that I can 'integrate both sides here' when I am integrating one side with respect to $x$ yet I ... |
H: In which interval there is a solution by Bolzano theorem?
Let a function of domain equal to $\mathbb{R}$ be $f(x)=e^x-3$.
In which of the follows intervals, by the Bolzano theorem, we can say that $f(x)=-x-\frac{3}{2}$ have at least one solution?
$A) \left ]0,\frac{1}{5} \right[$
$B) \left ]\frac{1}{5},\frac{1}{4... |
H: Why Zariski topology?
Why in algebraic geometry we usually consider the Zariski topology on $\mathbb A^n_k$? Ultimately it seems a not very interesting topology, infact the open sets are very large and it doesn't satisfy the Hausdorff separation axiom. Ok the basis is very simple, but what are the advantages?
AI: ... |
H: Understanding a Markov Chain
I am using a Markov Chain to get the 10 best search results from the union of 3 different search engines. The top 10 results are taken from each engine to form a set of 30 results.
The chain starts at State x, a uniform distribution of set S = {1,2,3,...30}. If the current state is page... |
H: Triples of positive real numbers $(a,b,c)$ such that $\lfloor a\rfloor bc=3,\; a\lfloor b\rfloor c=4,\;ab\lfloor c\rfloor=5$
Find the all ordered triplets of positive real numbers $(a,b,c)$ such that: $$\lfloor a\rfloor bc=3,\quad a\lfloor b\rfloor c=4,\quad ab\lfloor c\rfloor=5,$$
where $\lfloor x\rfloor$ is the ... |
H: Prove Axiom $10$ (Vector Spaces) independent of the others
Possible Duplicate:
Is it possible to construct a quasi-vectorial space without an identity element?
In Apostol Multivariable Calculus, $1.5$ exercise $30 b$, he asks the reader to prove that Axiom $10$ is independent of the other axioms for vector space... |
H: If $\sum a_n z^n = f(z) = \sum b_n z^n$, What Can Be Said About the Coefficients $a_n$ and $b_n$
I apologize for the cryptic title, this issue came up while obtaining a solution to my question here. I was given a power series:
$$\sum_{n=0}^{\infty}r(n)z^n = \frac{1}{(1-z)(1-z^2)(1-z^3)}$$
I was then asked to perf... |
H: At which points tangent to a curve is parallel to given plane?
At which points on the curve $\alpha(t):=(3t-t^3,3t^2,3t+t^3)$ the corresponding tangent lines are parallel to the plane $3x+y+z+2=0$?
AI: Hints:
The tangent line at $t$ has direction $\alpha'(t)$
The normal of a plane given by $ax + by + cz + d = 0$ i... |
H: How many $1$'s could there be in this sequence? Matrix, operator?
For each $(i,j)\in \mathbb{N}^2$, $a(i,j)=1$ or $0$, and 1) $a(i,i)=0$ for all $i$; 2)for fixed $i$, there is at most one $j$ such that $a(i,j)=1$. Suppose we know that there is a finite $\kappa$ such that
\begin{equation}
\sum_{i\in S, j\in S^{C}}a(... |
H: Upper-tail inequality for t-distribution
I am interested in upper tail bounds (or bounds on deviation from the mean) for t-distribution with n degrees of freedom (http://en.wikipedia.org/wiki/Student's_t-distribution)
A bound that is of the form similar to the tail bound for standard normal distribution
Proof of u... |
H: Learning Model Theory
What books/notes should one read to learn model theory? As I do not have much background in logic it would be ideal if such a reference does not assume much background in logic. Also, as I am interested in arithmetic geometry, is there a reference with a view towards such a topic?
AI: I really... |
H: Trying to reverse engineer this pattern...
this is my first post on the mathematics node of stack exchange, so please forgive me if I'm not posting an appropriate question, but I'm not sure where else to address this. I'm trying to figure out what equation might generate the following pattern. (keep in mind the num... |
H: Proving a commutative ring can be embedded in any quotient ring.
Here's the exercise, as quoted from B.L. van der Waerden's Algebra,
Show that any commutative ring $\mathfrak{R}$ (with or without a zero divisor) can be embedded in a ''quotient ring" consisting of all quotients $a/b$, with $b$ not a divisor of zero... |
H: To show the function $\frac{1}{x\log x}$ is continuous on $[2,\infty)$
I want to know that how can i show that the function $\displaystyle f(x)= \frac{1}{x\log x}$ continuous? Thanks in advance!
AI: If $f$ and $g$ are continuous on a particular interval and $g(x) \neq 0$ for any $x$ on that interval, then $\frac{f(... |
H: For any $n \geq 1$, prove that there exists a prime $p$ with at least n of its digits equal to $0$
For any $n \geq 1$, prove that there exists a prime $p$ with at least n of its digits
equal to $0$.
I don't even know how to start??
Any help(even a hint) would do.
Thanks in advance!!
AI: The series $\sum_{p \in \ma... |
H: Is this right? $ | \nabla f |^k \leqslant \sum_{i=1}^n | \partial_i f |^k $
Is this right if $ k \geqslant 1$ ? Then why?
$$ | \nabla f |^k \leqslant \sum_{i=1}^n | \partial_i f |^k \; $$
AI: This is essentially just the triangle equality for the so called $p$-norm (in your case $p=k$) together with the monotonici... |
H: Question on soluble groups
I'd like to ask for a clarification. I came across the following:
Lemma 2. Let $G$ be a finite solvable group and let $p$ be a prime number dividing $|G|$. Suppose that $M_1$ and $M_2$ are inconjugate maximal subgroups of $G$, both of which have $p$-power index in $G$, and neither of wh... |
H: How to prove this, $ | \Delta f | \leqslant n | \nabla^2 f| $
I hope to prove this,
$$ | \Delta f | \leqslant n | \nabla^2 f| $$
where $f : \mathbb R^n \to \mathbb R $.
AI: Use Cauchy-Schwarz inequality
$$
|\Delta f|=\left|\sum\limits_{i=1}^n \partial_i^2 f\right|=
\left|\sum\limits_{i=1}^n 1\cdot\partial_i^2 f\rig... |
H: $\lim\limits_{x\to\infty}x^a\sin(1/x)$
$$\lim_{x \rightarrow \infty} x^a\sin{\frac{1}{x}}$$
for this limit ,it was showed on the textbook that
$$\lim_{x \rightarrow \infty} x^a\sin{\frac{1}{x}}=\begin{cases} 0 & a<1 \\ 1 & a=1 \\ \infty & a>1 \end{cases}$$
in my opinion ,it's obviously that $\lim\limits_{x \rightar... |
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